Questions concerning Decoherence and Entanglement

Hello, I have two questions I wish to ask concerning Decoherence and entanglement:

1. I am certainly no expert on quantum mechanics, and while I was reading I stumbled upon the concept of decoherence. I understand the idea, but I have a few questions concerning it:

1. theoretically, if the environment "slows down," similar to entering a state of low energy, can a particle that is entangled still produce interferences?
2. Assuming that a system is always changing in some way, does interference completely cancel out?
3. Do particle interferences arise faster than the speed of light (assuming the entangled system is not dissipative)?

I also have a few questions concerning entanglement:

1. Is entanglement permanent within a system?
2. Is everything in the universe entangled?
3. If the above is true, then why are there are so much differences between the states of all systems?
4. How and which properties are usually correlated?
5. Is there "anti-entanglement"?

I wouldn't know if I don't know the answers to your questions, because I don't understand your questions in the first place. For example, what is "environment "slows down," similar to entering a state of low energy"? Or what about this: "Assuming that a system is always changing in some way"? What property exactly that is "changing in some way"?

One of the things we learn in physics is that the questions that we asked must be clearly defined, or else we would not know what to look for. Concepts and ideas in physics have clearly, underlying mathematical definition. This means that in particular cases, there are certain well-defined description of things such as "entanglement", "energy of the system", etc.. When you use that in a mix-and-match way without knowing the "rules" in which they can be used, then you can easily end up with something that do not make sense or has no clear definition. Questions like these typically either have no answers, or have varying answers depending on how a reader interprets the question. This means that you'll end up with a bunch of different answers based on different premises, and that can only mean a jumbled mess.

I'm guessing that most people on here have encountered the latter scenario on here and simply have no "energy" to be involved in another one. I know I am.

I wouldn't know if I don't know the answers to your questions, because I don't understand your questions in the first place. For example, what is "environment "slows down," similar to entering a state of low energy"? Or what about this: "Assuming that a system is always changing in some way"? What property exactly that is "changing in some way"?

One of the things we learn in physics is that the questions that we asked must be clearly defined, or else we would not know what to look for. Concepts and ideas in physics have clearly, underlying mathematical definition. This means that in particular cases, there are certain well-defined description of things such as "entanglement", "energy of the system", etc.. When you use that in a mix-and-match way without knowing the "rules" in which they can be used, then you can easily end up with something that do not make sense or has no clear definition. Questions like these typically either have no answers, or have varying answers depending on how a reader interprets the question. This means that you'll end up with a bunch of different answers based on different premises, and that can only mean a jumbled mess.

I'm guessing that most people on here have encountered the latter scenario on here and simply have no "energy" to be involved in another one. I know I am.

Zz.

What I meant by the "environment "slows down", similar to a low state of energy" question is if the surroundings of an entangled system enter lower states of energy (for whatever reason), will a particle in this system start 'producing' interferences? I thought that decoherence was an effect only if the entangled system is continuously in motion.

By "changing in some way" I meant that: If a entangled system exhibited change in any way, (any variable from spin to energy), is there any possibility of interferences manifesting?

So, are you suggesting there is no paradigm that describes decoherence or that my langauge in expressing my questions is too ambiguous? I apologize if my language did not make sense earlier.

What I meant by the "environment "slows down", similar to a low state of energy" question is if the surroundings of an entangled system enter lower states of energy (for whatever reason), will a particle in this system start 'producing' interferences? I thought that decoherence was an effect only if the entangled system is continuously in motion.

Ok.. if I have 2 entangled photons as in a typical EPR-type experiment, can you indicate to me where would the interaction with the "environment" comes in here?

By "changing in some way" I meant that: If a entangled system exhibited change in any way, (any variable from spin to energy), is there any possibility of interferences manifesting?

First of all, what "interferences" are you alluding to? Remember that most entanglement demonstrations do not make any "intereferences", but rather measures correlations. The only experiment that comes close to measuring "interferences" with entangled particles are the recent experiment that showed that entangled photons can beat the diffraction limit. However, I do not think this is what you are refering to. So can you describe the scenario of such "interference"?

So, are you suggesting there is no paradigm that describes decoherence or that my langauge in expressing my questions is too ambiguous? I apologize if my language did not make sense earlier.

There have been several studies that describe decoherence of a quantum mechanical system, but I am not aware of any yet that connects decoherence with entanglement. You need to keep in mind that the reason that we do not encounter entanglement that easily IS due to decoherence. It can easily (and very often, does) destroy the entanglement between particles.

Ok.. if I have 2 entangled photons as in a typical EPR-type experiment, can you indicate to me where would the interaction with the "environment" comes in here?

First of all, what "interferences" are you alluding to? Remember that most entanglement demonstrations do not make any "intereferences", but rather measures correlations. The only experiment that comes close to measuring "interferences" with entangled particles are the recent experiment that showed that entangled photons can beat the diffraction limit. However, I do not think this is what you are refering to. So can you describe the scenario of such "interference"?

There have been several studies that describe decoherence of a quantum mechanical system, but I am not aware of any yet that connects decoherence with entanglement. You need to keep in mind that the reason that we do not encounter entanglement that easily IS due to decoherence. It can easily (and very often, does) destroy the entanglement between particles.

Zz.

To the first comment, I was referring to, say, a larger entanglement system. What I mean by 'interferences' is the wave properties a particle exhibits which 'arise' naturally. So, assuming an entangled system, take the other particle in the EPR experiment, is always acting upon the other (which is something I am assuming to be true at this point), interferences of the original particle should never arise correct? But, from what I have read, wave interferences do. So, does entanglement only have an effect when the other particle is dynamic, and affecting the other particle with any change?

I don't know anything about decoherence destroying entanglement, as such, I thought entanglement is something that was forever standing. Could you explain? Thanks.

To the first comment, I was referring to, say, a larger entanglement system. What I mean by 'interferences' is the wave properties a particle exhibits which 'arise' naturally. So, assuming an entangled system, take the other particle in the EPR experiment, is always acting upon the other (which is something I am assuming to be true at this point), interferences of the original particle should never arise correct? But, from what I have read, wave interferences do. So, does entanglement only have an effect when the other particle is dynamic, and affecting the other particle with any change?

I don't know anything about decoherence destroying entanglement, as such, I thought entanglement is something that was forever standing. Could you explain? Thanks.

I'm sorry, but I think you've lost me here. Maybe someone else understands this better than I do.

There have been several studies that describe decoherence of a quantum mechanical system, but I am not aware of any yet that connects decoherence with entanglement. You need to keep in mind that the reason that we do not encounter entanglement that easily IS due to decoherence. It can easily (and very often, does) destroy the entanglement between particles.

I can comment on this one: decoherence IS in fact "wild" entanglement with the environment, which is practically irreversible. Remember that "entanglement" is only visible when we look at CORRELATIONS between measurements on the two entangled systems. In the particular case of EPR, for instance, the Alice and Bob photons, *when looked at individually*, behave like a statistical mixture and not a superposition. The superposition (the quantum interference effects, distinguishing a superposition from a statistical mixture) are ONLY visible in *correlations* between measurements on the two photons. As such, a pair of entangled photons looks "less quantum-mechanical" than a single photon beam, which can produce local interference effects. Locally, the beams at Alice and at Bob are "white" so to speak, and don't really show as much interference as a "pure" beam. But such an EPR pair is special, in that the entanglement is still limited to just a pair, and that we still have control over ALL THE COMPONENTS OF THE ENTANGLED SYSTEM.

If we look at an entangled threesome, then we the quantum interference effects are only visible in the 3rd order correlation functions between measurements on the 3 components: the individual measurements look like those of mixtures, and so do the second order correlations: they look like statistical mixtures.

Now, with entanglement with the environment, we've LOST the control over all the components of the system, as they are myriads, and of different nature. So, the entanglement with the environment leads us to see a system as just a statistical mixture, with no interference effects (limited to the system) left.
As such, the entanglement with the environment has the effect upon the system, locally, of suppressing all forms of observable interference. As with the EPR pair, one should *in principle* be able to find "strange correlations" between measurements on the system and on ALL the entangled components of the environment, but these observations are practically impossible. So we NEVER see such "strange correlations", and conclude that the quantum superposition has been transformed into a statistical mixture (but that's only because the correlation has now been promoted to such high-order and unmeasurable correlation function, that we never notice).

So "uncontrolled entanglement" promotes quantum interference to high-n correlation functions which are totally unobservable for all practical purposes.

The individual systems which get entangled with the environment loose hence all form of "coherence" by themselves.

I can comment on this one: decoherence IS in fact "wild" entanglement with the environment, which is practically irreversible. Remember that "entanglement" is only visible when we look at CORRELATIONS between measurements on the two entangled systems. In the particular case of EPR, for instance, the Alice and Bob photons, *when looked at individually*, behave like a statistical mixture and not a superposition. The superposition (the quantum interference effects, distinguishing a superposition from a statistical mixture) are ONLY visible in *correlations* between measurements on the two photons. As such, a pair of entangled photons looks "less quantum-mechanical" than a single photon beam, which can produce local interference effects. Locally, the beams at Alice and at Bob are "white" so to speak, and don't really show as much interference as a "pure" beam. But such an EPR pair is special, in that the entanglement is still limited to just a pair, and that we still have control over ALL THE COMPONENTS OF THE ENTANGLED SYSTEM.

I don't know what you meant by the beam, but whatever. Makes enough sense to me.

If we look at an entangled threesome, then we the quantum interference effects are only visible in the 3rd order correlation functions between measurements on the 3 components: the individual measurements look like those of mixtures, and so do the second order correlations: they look like statistical mixtures.

Now, with entanglement with the environment, we've LOST the control over all the components of the system, as they are myriads, and of different nature. So, the entanglement with the environment leads us to see a system as just a statistical mixture, with no interference effects (limited to the system) left.
As such, the entanglement with the environment has the effect upon the system, locally, of suppressing all forms of observable interference. As with the EPR pair, one should *in principle* be able to find "strange correlations" between measurements on the system and on ALL the entangled components of the environment, but these observations are practically impossible. So we NEVER see such "strange correlations", and conclude that the quantum superposition has been transformed into a statistical mixture (but that's only because the correlation has now been promoted to such high-order and unmeasurable correlation function, that we never notice).

So "uncontrolled entanglement" promotes quantum interference to high-n correlation functions which are totally unobservable for all practical purposes.

The individual systems which get entangled with the environment loose hence all form of "coherence" by themselves.

So, it seems as if larger entangled systems are impossible to determine correlations am I correct?

I don't really know what you mean by statistical mixture.

So, what how it comes to me, it seems like you are stating that entangled systems DO NOT correlate observables but, for some reason, interference doesn't exist. Isn't this the opposite of entanglement, if pieces of a system just fail to make correlation with each other? Also, is there any way to figure out why systems collapse superposition if there is no entanglement in the sense of the EPR pair?

Thanks for answering the first of my questions. I hope to see more answers too.

I don't know what you meant by the beam, but whatever. Makes enough sense to me.

Beam of photons...

For a pure beam, all photons are in a single quantum state (a vector in hilbert space). For a mixture, they are, well, coming in a statistical mixture of different pure states (although this expression, by itself, needs some caveats - but not all difficulties at once )

The point is that observational differences between different "states" are only visible when doing statistical measurements on a big number of "identical" systems. So we're talking here about the observational difference between the components of an entangled system (read: on a whole series of such systems, or a beam of such particles in this case), and those that aren't.

So, it seems as if larger entangled systems are impossible to determine correlations am I correct?

The bigger the number of entangled components, the harder it is to find correlations between observations which are different from those of a classical statistical mixture BUT how more striking and puzzling they are when they are observed!

I don't really know what you mean by statistical mixture.

Several specimen in different states, arriving one after the other at the measurement, randomly mixed.

Quantum theory's only reason of existence is that there are states in nature which seem to be DIFFERENT from simple statistical mixtures, so the "quantum-ness" of an observation is the difference between such mixture and the quantum predictions.

In short: there's a difference between the quantum state:
|psi> = |a> + |b> - which is a pure state on one hand (sheer "quantumness")
and:
a statistical mixture of 50% of things coming in in state |a> and 50% of things coming in in state |b>.

But you only see the difference if you do 2 things:
1) you do observations on MANY of these "identical" systems
2) you look at the right quantities. For instance: if you look at a property which is determined by state |a> or by state |b> (in other words, if |a> and |b> are eigenvectors of the measurement operator), both cases 1) and 2) will give identical results. IOW, we haven't seen any "quantum effect" when doing that. However, if you look at a quantity which is determined by |c> and |d>, where |c> = |a> + |b> and |d> = |a> - |b>, you WILL see a difference: in case 1), in 100% of the cases, you will see the c-property and never the d-property ; while in case 2), you will find 50% of c-property and 50% of d-property.

It is in this kind of case, where you find a difference between a pure state and a statistical mixture, that you can say you have observed a "quantum effect" or "quantum interference" or something of the kind.

Well, in the case of entangled systems, these observations showing such effects need to be measurements on ALL components of the entangled system: if you miss one, it turns out like if it were a statistical mixture. IF you observe them, they are very puzzling. But if the entanglement is too complicated, you always leave out one necessary measurement on some part of the system, and hence you don't see any quantum effect: everything behaves as a mixture. So although there "are" very puzzling quantum effects to be potentially observed, you can never actually do so when there is entanglement with the environment ; and hence things appear to be "just statistical mixtures" with no quantum effects per se. This is the essential idea of decoherence theory.

So, what how it comes to me, it seems like you are stating that entangled systems DO NOT correlate observables but, for some reason, interference doesn't exist.

No, that's not what I'm saying. I'm saying that in entangled systems, one needs observables which observe ALL of the components before we can find specific "quantum correlations". If we leave one out, we won't see it. And in a complicated system, we will almost always miss one, so we'll never OBSERVE the correlations.

Isn't this the opposite of entanglement, if pieces of a system just fail to make correlation with each other? Also, is there any way to figure out why systems collapse superposition if there is no entanglement in the sense of the EPR pair?

There IS (according to decoherence) entanglement in the sense of the EPR pair, only, one part of the pair is unobservable. And if you only look at ONE PART of the EPR pair, it doesn't look particularly correlated with anything: it shows up as a mixture.
Now, in the case of an EPR pair, we can go and do observations on the single other partner in the entanglement, and find amazing correlations. But if it is not a pair, but a billion-some, then there will always be one partner that escapes observation. And it is only on the total set of observations that a correlation (an amazing correlation) is visible. On any subset of observations, the entangled state appears as a mixture.

For a pure beam, all photons are in a single quantum state (a vector in hilbert space). For a mixture, they are, well, coming in a statistical mixture of different pure states (although this expression, by itself, needs some caveats - but not all difficulties at once )

The point is that observational differences between different "states" are only visible when doing statistical measurements on a big number of "identical" systems. So we're talking here about the observational difference between the components of an entangled system (read: on a whole series of such systems, or a beam of such particles in this case), and those that aren't.

The bigger the number of entangled components, the harder it is to find correlations between observations which are different from those of a classical statistical mixture BUT how more striking and puzzling they are when they are observed!

Several specimen in different states, arriving one after the other at the measurement, randomly mixed.

Quantum theory's only reason of existence is that there are states in nature which seem to be DIFFERENT from simple statistical mixtures, so the "quantum-ness" of an observation is the difference between such mixture and the quantum predictions.

In short: there's a difference between the quantum state:
|psi> = |a> + |b> - which is a pure state on one hand (sheer "quantumness")
and:
a statistical mixture of 50% of things coming in in state |a> and 50% of things coming in in state |b>.

But you only see the difference if you do 2 things:
1) you do observations on MANY of these "identical" systems
2) you look at the right quantities. For instance: if you look at a property which is determined by state |a> or by state |b> (in other words, if |a> and |b> are eigenvectors of the measurement operator), both cases 1) and 2) will give identical results. IOW, we haven't seen any "quantum effect" when doing that. However, if you look at a quantity which is determined by |c> and |d>, where |c> = |a> + |b> and |d> = |a> - |b>, you WILL see a difference: in case 1), in 100% of the cases, you will see the c-property and never the d-property ; while in case 2), you will find 50% of c-property and 50% of d-property.

It is in this kind of case, where you find a difference between a pure state and a statistical mixture, that you can say you have observed a "quantum effect" or "quantum interference" or something of the kind.

Well, in the case of entangled systems, these observations showing such effects need to be measurements on ALL components of the entangled system: if you miss one, it turns out like if it were a statistical mixture. IF you observe them, they are very puzzling. But if the entanglement is too complicated, you always leave out one necessary measurement on some part of the system, and hence you don't see any quantum effect: everything behaves as a mixture. So although there "are" very puzzling quantum effects to be potentially observed, you can never actually do so when there is entanglement with the environment ; and hence things appear to be "just statistical mixtures" with no quantum effects per se. This is the essential idea of decoherence theory.

No, that's not what I'm saying. I'm saying that in entangled systems, one needs observables which observe ALL of the components before we can find specific "quantum correlations". If we leave one out, we won't see it. And in a complicated system, we will almost always miss one, so we'll never OBSERVE the correlations.

There IS (according to decoherence) entanglement in the sense of the EPR pair, only, one part of the pair is unobservable. And if you only look at ONE PART of the EPR pair, it doesn't look particularly correlated with anything: it shows up as a mixture.
Now, in the case of an EPR pair, we can go and do observations on the single other partner in the entanglement, and find amazing correlations. But if it is not a pair, but a billion-some, then there will always be one partner that escapes observation. And it is only on the total set of observations that a correlation (an amazing correlation) is visible. On any subset of observations, the entangled state appears as a mixture.

I suppose that answers some of my questions. Thanks. Another I wish to ask is: Is entanglement permanent, or is it a temporary state of stronger uniformity? And: Why is it that when you measure all aspects of an entangled system you get 'quantum effects'? Classical logic tells me the opposite should occur...

I suppose that answers some of my questions. Thanks. Another I wish to ask is: Is entanglement permanent, or is it a temporary state of stronger uniformity? And: Why is it that when you measure all aspects of an entangled system you get 'quantum effects'? Classical logic tells me the opposite should occur...

Sorry, this was my other account which I logged into by mistake (don't ask)...

I suppose that answers some of my questions. Thanks. Another I wish to ask is: Is entanglement permanent, or is it a temporary state of stronger uniformity?

This is a matter of interpretation. I, for one, think that the best way to look upon quantum theory is through the so-called "many-worlds" view, and in that case, entanglement is permanent. Others, which adhere in one way or another to a "collapse" view, would say that entanglement only lasts up to collapse.

And: Why is it that when you measure all aspects of an entangled system you get 'quantum effects'? Classical logic tells me the opposite should occur...

You get "quantum effects" each time that you measure a pure state of a quantum system in a different basis than its "intuitive" basis. For instance, in the two-slit experiment, the "intuitive basis" is "goes through left slit" and "goes through right slit", and the measurement basis is "hits position x on the screen". If we take it that the intuitive basis is given by a statistical mixture (some go through the left slit, others go through the right slit), then we are surprised to not find this mixture again in our distribution of x-position hits on the screen (two bumps), but something which differs from it: an "interference pattern". The "interference pattern" is exactly what is the difference between the "mixture of the intuitive basis" prediction (two bumps), and the actual observed probability density (a wavy pattern).

So, for pure states of single particles, the "quantum effect" already resides in an interference pattern (which is the difference between the actual probability density observed/predicted, and that which should have resulted from the pure application of the statistical mixture of the "intuitive" basis).

This always happens when the observed state is a pure state in superposition of "intuitive basis states", and when we look at an observation which is NOT that same basis. This comes simply about because of the "absolute square" rule of complex numbers in quantum theory: the fact that if u and v are complex numbers, that |u+v|^2 = |u|^2 + |v|^2 + 2 Re(u.v*)

The first two terms is what we obtain also in the the "statistical mixture of the intuitive basis" view, and the last term is the "interference term" which gives us the difference with the quantum prediction. It's all in this last term, and it is THIS term, in all circumstances, which is the entire content of "quantum effects". It is this term which gives us the "interference patterns" in the two-slit experiment.

In the EPR setup, the pure state is |u>|d> - |d>|u>. The "intuitive basis" is one in which each particle "has its own state", hence spanned by:
|u>|u> ; |u>|d> ; |d>|u> and |d>|d>, and our pure state is clearly a superposition of these "individual particle state" states. THIS kind of superposition is called an "entangled state", btw: when "individual states of the subsystems" are considered to be the "intuitive basis", and when the state is not in just one of those.

So in this case, we will only see possible "interference effects" as compared to a statistical mixture of "intuitive basis states", when we do a measurement which is NOT one with an eigenbasis equivalent to the "intuitive basis" ; in other words, it will need to be a correlation measurement, which has eigenstates NOT of the kind |u>|d>,...
Measurements ONLY affecting one subsystem WILL have such an eigenbasis corresponding to the intuitive basis, and hence in those measurements, the "quantum effects" will not show up.

This is a matter of interpretation. I, for one, think that the best way to look upon quantum theory is through the so-called "many-worlds" view, and in that case, entanglement is permanent. Others, which adhere in one way or another to a "collapse" view, would say that entanglement only lasts up to collapse.

You get "quantum effects" each time that you measure a pure state of a quantum system in a different basis than its "intuitive" basis. For instance, in the two-slit experiment, the "intuitive basis" is "goes through left slit" and "goes through right slit", and the measurement basis is "hits position x on the screen". If we take it that the intuitive basis is given by a statistical mixture (some go through the left slit, others go through the right slit), then we are surprised to not find this mixture again in our distribution of x-position hits on the screen (two bumps), but something which differs from it: an "interference pattern". The "interference pattern" is exactly what is the difference between the "mixture of the intuitive basis" prediction (two bumps), and the actual observed probability density (a wavy pattern).

So, for pure states of single particles, the "quantum effect" already resides in an interference pattern (which is the difference between the actual probability density observed/predicted, and that which should have resulted from the pure application of the statistical mixture of the "intuitive" basis).

This always happens when the observed state is a pure state in superposition of "intuitive basis states", and when we look at an observation which is NOT that same basis. This comes simply about because of the "absolute square" rule of complex numbers in quantum theory: the fact that if u and v are complex numbers, that |u+v|^2 = |u|^2 + |v|^2 + 2 Re(u.v*)

The first two terms is what we obtain also in the the "statistical mixture of the intuitive basis" view, and the last term is the "interference term" which gives us the difference with the quantum prediction. It's all in this last term, and it is THIS term, in all circumstances, which is the entire content of "quantum effects". It is this term which gives us the "interference patterns" in the two-slit experiment.

In the EPR setup, the pure state is |u>|d> - |d>|u>. The "intuitive basis" is one in which each particle "has its own state", hence spanned by:
|u>|u> ; |u>|d> ; |d>|u> and |d>|d>, and our pure state is clearly a superposition of these "individual particle state" states. THIS kind of superposition is called an "entangled state", btw: when "individual states of the subsystems" are considered to be the "intuitive basis", and when the state is not in just one of those.

So in this case, we will only see possible "interference effects" as compared to a statistical mixture of "intuitive basis states", when we do a measurement which is NOT one with an eigenbasis equivalent to the "intuitive basis" ; in other words, it will need to be a correlation measurement, which has eigenstates NOT of the kind |u>|d>,...
Measurements ONLY affecting one subsystem WILL have such an eigenbasis corresponding to the intuitive basis, and hence in those measurements, the "quantum effects" will not show up.

Can you suggest a type of measurement that is not eigenbased?

Also, if subsystems' quantum effects do not show up, then why do they 'appear' in, say, the double slit experiment?

I think I am missing something with interference, outside of its definition of wave-superposition. To me, you seem to be speaking of a different kind of superposition. Could you continue to explain on this? Thanks again.

With every measurement corresponds an eigenbasis, by definition. So I fail to see what you mean.

Also, if subsystems' quantum effects do not show up, then why do they 'appear' in, say, the double slit experiment?

I suppose that you mean: given that everything gets finally entangled with everything, how come that we can still see some quantum effects somewhere ?

The point is that in order to see quantum effects, you have to "prepare" a subsystem, followed by a "measurement". If you do raw measurements on a subsystem which is entangled with other stuff, you will not see any "interference" at all, because the subsystem will appear to you as being part of a statistical mixture. However, if you prepare (filter !) a subsystem, you will be able again, to see interference. But the preparation is ALSO a kind of measurement! As such, the observed quantum effect is nothing else but a correlation between two successive measurements (the preparation, followed by the actual measurement). If, between both, you do not have any interactions which might "decohere" (read: entangle) your subsystem with anything else, then these correlations may show up.

With every measurement corresponds an eigenbasis, by definition. So I fail to see what you mean.

I suppose that you mean: given that everything gets finally entangled with everything, how come that we can still see some quantum effects somewhere ?

The point is that in order to see quantum effects, you have to "prepare" a subsystem, followed by a "measurement". If you do raw measurements on a subsystem which is entangled with other stuff, you will not see any "interference" at all, because the subsystem will appear to you as being part of a statistical mixture. However, if you prepare (filter !) a subsystem, you will be able again, to see interference. But the preparation is ALSO a kind of measurement! As such, the observed quantum effect is nothing else but a correlation between two successive measurements (the preparation, followed by the actual measurement). If, between both, you do not have any interactions which might "decohere" (read: entangle) your subsystem with anything else, then these correlations may show up.

Relating to the above paragraph, can you explain?

One thing I am understanding is the idea of having a 'pure state' being a different state that is an 'interference pattern.' If this is so, is the eigenstate not a 'pure' state, or is that just a consequence of the HUP, that as more information is known the lesser-known properties become more 'unknown'?

And, in relation to this statement: "
In the EPR setup, the pure state is |u>|d> - |d>|u>. The "intuitive basis" is one in which each particle "has its own state", hence spanned by:
|u>|u> ; |u>|d> ; |d>|u> and |d>|d>, and our pure state is clearly a superposition of these "individual particle state" states. THIS kind of superposition is called an "entangled state", btw: when "individual states of the subsystems" are considered to be the "intuitive basis", and when the state is not in just one of those."

Does that mean that entanglement is nothing more than the 'strange state' or the term 'entangled state' just part of the terminology used. It is because of uses of the term 'entanglement' that leads me to suspect that it is something a bit different from "correlating observables." It's probably my ego, but can you clarify this?

So, what I have gotten so far is that measuring any system with a 'intuitive basis,' finding certain eigenvalues of the system, will result in quantum interferences of some kind. And when we don't measure something, the system is in a superposition of 'intuitive states.'

Finally, can you give a different example of 'interferences,' because my mind is confused about what they really are in a quantum system outside of a 'wave-patter' in the double slit experiment.

One thing I am understanding is the idea of having a 'pure state' being a different state that is an 'interference pattern.' If this is so, is the eigenstate not a 'pure' state, or is that just a consequence of the HUP, that as more information is known the lesser-known properties become more 'unknown'?

mmm. The problem I have in this discussion, is that I fail to see the way you picture things. This is a necessary condition in order for me to try to find the "right" way of explaining things. I really cannot make anything of what you write above. For instance, when you write: "the idea of having a pure state being A DIFFERENT STATE than is an interference pattern". This would imply somehow that "an interference pattern" is a state, which it isn't: it is a result of a measurement.
An "eigenstate" is of course always a pure state, and I really really don't see what the HUP is doing in this.

So, given that I cannot understand exactly what you ask and how you see things, I have no way of trying to give an explanation that might make any sense to you. I can try to write some elementary statements, but I don't know if they are related to what you are asking/saying.

1) "pure states" are quantum states which are vectors in hilbert space. We talk about an individual specimen, but we think in fact about an entire beam of such specimen. They are such, that there exists a complete set of observables (= a set of compatible measurements) for which ALL systems in that beam give exactly the same results to all of these measurements. There is, in other words, no statistical spread in the outcomes, and the system behaves completely deterministically. Mind you that you don't have the choice about WHICH observables to pick.

For instance, a particle can be in a pure state, which happens to be a position state. A beam of such particles will then yield ALWAYS THE SAME RESULT when we do position measurements X, Y and Z on them.
But a particle can be also in another pure state, which happens to be a momentum state. In that case, a beam of such particles will ALWAYS GIVE THE SAME OUTCOMES when we do a momentum measurement (but not when we do a position measurement!). So we see that a pure state is somehow associated with a set of observables for which the outcomes will be determined with certainty, but it is the state which determines the observables. We don't have the choice. For most pure states, however, these observables are only theoretical, and are not really realizable as a measurement in the lab (although they could, in principle).
The set of all these possible pure states span the hilbert space of quantum states.

But we can also think of a beam of particles, of which not all of them are in the same pure state, but which are statistically mixed. One might be in a position state , the next might be in a momentum state, etc...
In such a case, we say that the beam (and by extrapolation, each individual in the beam) is "in a mixture".
A beam in a mixture is such, that there doesn't exist, even in theory, any complete set of observables for which the outcome is always the same. We will ALWAYS have a statistical spread of outcomes, no matter what kind of measurement we do. This wasn't the case for a beam in a pure state: there, there existed at least ONE COMPLETE SET of measurements for which the outcomes would always be the same.

However, if we apply, to a beam in a pure state, a set of measurements, which is not the "good" set, then we have ALSO a statistical distribution of outcomes. So if we limit us to such measurements, we're not really making a difference between "a statistical mixture" and "a pure state".

Quantum effects typically show up when:
1) in an "intuitive set of measurements" we seem to have a mixture
2) in a specific set of measurements which are not so intuitive, we "always find the same result".

Because of 1), one would be tempted to think of the beam as "just being a statistical mixture of stuff", and then 2) is entirely puzzling, because 2) cannot happen for a GENUINE statistical mixture of stuff. Almost all (if not all) "paradoxes" in quantum theory can be reduced to such a scheme.

Does that mean that entanglement is nothing more than the 'strange state' or the term 'entangled state' just part of the terminology used. It is because of uses of the term 'entanglement' that leads me to suspect that it is something a bit different from "correlating observables." It's probably my ego, but can you clarify this?

Entanglement is a specific case of "non-intuitive pure states". Entanglement is that set of pure states, when we look at PURE quantum states of systems which consist of (spatially separated) SUBSYSTEMS. That means that *intuitively* we would be tempted to assign individual states to the subsystems, as we think of them as "separated". But mathematically, if we assign a specific state |u> to system 1 and a state |a> to system 2, then the overall state is of the kind |u>|a>. Now, NOT ALL PURE STATES of the combined system can be written in that form ; in other words, we've severely limited the INTUITIVE set of states, and the actual set of pure states is quite larger. All states that are pure states, but NOT of the kind |state 1> |state 2>, are called ENTANGLED states.

The point is that, because they are nevertheless pure states, that there are "funny outcomes" of certain measurements, which are not compatible with the mixture we might think is there, if we look upon it in the "intuitive basis".

So, what I have gotten so far is that measuring any system with a 'intuitive basis,' finding certain eigenvalues of the system, will result in quantum interferences of some kind. And when we don't measure something, the system is in a superposition of 'intuitive states.'

Yes.

Finally, can you give a different example of 'interferences,' because my mind is confused about what they really are in a quantum system outside of a 'wave-patter' in the double slit experiment.

mmm. The problem I have in this discussion, is that I fail to see the way you picture things. This is a necessary condition in order for me to try to find the "right" way of explaining things. I really cannot make anything of what you write above. For instance, when you write: "the idea of having a pure state being A DIFFERENT STATE than is an interference pattern". This would imply somehow that "an interference pattern" is a state, which it isn't: it is a result of a measurement.
An "eigenstate" is of course always a pure state, and I really really don't see what the HUP is doing in this.

So, given that I cannot understand exactly what you ask and how you see things, I have no way of trying to give an explanation that might make any sense to you. I can try to write some elementary statements, but I don't know if they are related to what you are asking/saying.

I basically use a kind of logic that is a 'branch' from classical logic, considering that I am a neophyte to quantum physics but have read some non-mathematical books about it. Saying this, I suppose I picture things in quantum physics almost literally fundamentally, until someone clarifies the exact way one to think about a given concept of quantum mechanics. For instance, when I first read about wave-particle duality, I thought that an electron was a particle traveling along a sinusoidal path. Naturally, as I read more about the topic, a 'clearer picture came to.' So, when I first read about decoherence and entanglement, which is very recent, I hard a difficult time visualizing such a concept. Of course, you're explanation seems to have helped a lot, and I am grateful.

vanesch said:

Quantum effects typically show up when:
1) in an "intuitive set of measurements" we seem to have a mixture
2) in a specific set of measurements which are not so intuitive, we "always find the same result".

Because of 1), one would be tempted to think of the beam as "just being a statistical mixture of stuff", and then 2) is entirely puzzling, because 2) cannot happen for a GENUINE statistical mixture of stuff. Almost all (if not all) "paradoxes" in quantum theory can be reduced to such a scheme.

I still don't understand what a 'non-intuitive measurement' is; aren't all quantum measurements intuitive?

vanesch said:

Entanglement is a specific case of "non-intuitive pure states". Entanglement is that set of pure states, when we look at PURE quantum states of systems which consist of (spatially separated) SUBSYSTEMS. That means that *intuitively* we would be tempted to assign individual states to the subsystems, as we think of them as "separated". But mathematically, if we assign a specific state |u> to system 1 and a state |a> to system 2, then the overall state is of the kind |u>|a>. Now, NOT ALL PURE STATES of the combined system can be written in that form ; in other words, we've severely limited the INTUITIVE set of states, and the actual set of pure states is quite larger. All states that are pure states, but NOT of the kind |state 1> |state 2>, are called ENTANGLED states.

The point is that, because they are nevertheless pure states, that there are "funny outcomes" of certain measurements, which are not compatible with the mixture we might think is there, if we look upon it in the "intuitive basis".

I never would have thought of entanglement that way. Interesting view.

If I may ask one more question: how does this all tie into Bell's theorem and nonlocality/acasaulity? I'm just getting curious...

I still don't understand what a 'non-intuitive measurement' is; aren't all quantum measurements intuitive?

Not "non-intuive measurement", but measurement with a non-intuitive eigenbasis.

You have to know that the whole idea of quantum mechanics is the superposition principle. I keep stressing that over here ; it is its fundamental principle, on which all is build up. A bit like the fundamental idea of general relativity is that there is a spacetime manifold ; well, in the same way, the corner stone of quantum theory is the superposition principle.

Again, it is the cornerstone of the *theory*. We don't know if it is a fundamental principle in ALL OF NATURE, but quantum theory, as we know it today, as a theory, is build around this principle. I should add that, in all cases where we could kind of verify the specific consequences of the idea experimentally, it got confirmed. But usually, this was on "rather small" systems, so there might still be a violation of the principle in nature. We don't know. But for the moment, let's stick with "a world in which quantum theory as we know it, is correct".

Well, the superposition principle tells us the following (extremely strange!) thing: if a thing can be observed in a state A and a thing can be observed in a state B, well, then it can also exist in any kind of complex superposition c1 A + c2 B. Now, that's strange for sure ! If a train can be observed in London, and the train can be observed in Paris, then it can be also in a state which is a superposition of both
(trains are typically objects for which we ignore whether yes or no, the superposition principle applies ! I just do this for some drama in the exposition...)
Usually, we limit this kind of weird idea to smaller things, like electrons, or photons or the like. The electron can "go through the left slit" is in principle an observable state. The electron can "go through the right slit" also. Well, all thinkable complex superpositions of these two are also possible states.

But what is REALLY weird, is that these "states in superposition" can be *different states of observation*. Specific combinations of "states of a specific position" for instance, correspond to a *single specific outcome of another measurement, such as momentum*.

Now, there are "states" which look to us to be "far more intuitive" than others. For instance, the train being in a certain station, is far more intuitive than the "train being in a state which is a superposition of being in two stations", EVEN THOUGH there corresponds in principle, a specific observation which corresponds to exactly that. Now, for trains, I don't know what would be the specific measurement that would correspond to this state. But for electrons, we know: the superposition of several different "position states" corresponds to a single "momentum state" for instance.
The superposition of "spin up" and spin down" states, corresponds to a "spin on the left side" state.

Clearly, "position" is a such an "intuitive" state: we expect a particle to "have a specific position in space". Sometimes we can find however, measurable properties which correspond to superpositions of these intuitive states.
And it is when we can perform measurements that correspond to these "weird superpositions of intuitive states" that we can hope to see quantum effects.

If I may ask one more question: how does this all tie into Bell's theorem and nonlocality/acasaulity? I'm just getting curious...

As I said, entanglement occurs when we have "several subsystems". We can think of a system sliced up into different subsystems, and then an "intuitive" basis is when each subsystem has a specific state (it's difficult to imagine otherwise!). Well, entangled states are NOT such states, but superpositions thereof.

One thing I am failing to reconcile in my head is the idea of the entanglement seen in EPR. Accordingly, are things 'correlated' in nature, or are they only correlated when one wishes to observe correlations? Also, if entanglement is simply something being in a state like Schroedinger's cat experiment, then how does superposition fall into the manifestation of the second particle. In other words, why does the observed particle's observed eigenstate affect the other member of the EPR pair?

Also, if preparing measurement is a type of measurement, does that mean something as radical as stating that 'imaging' an experiment is a type of measurement? Also, I don't really how this is a type of measurement in the end...